报告题目：On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems

报 告 人：刁怀安 东北师范大学数学与统计学院副教授

报告时间：2019年7月23日13:30

报告地点：数学楼第一报告厅

Abstract：

In this talk, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviors of the eigenfunctions in two dimensions. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We establish an accurate and comprehensive quantitative characterization of the relationship. Roughly speaking, the vanishing order is generically infinite if the intersecting angle is irrational, and the vanishing order is finite if the intersecting angle is rational. In fact, in the latter case, the vanishing order is the degree of the rationality. The theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal setup that one can recover the support of the unknown scatter as well as the surface impedance parameter by finitely many far-field patterns. Indeed, at most two far-field pat- terns are sufficient for some important applications. Unique identifiability by finitely many far-field patterns remains to be a highly challenging fundamental mathematical problem in the inverse scattering theory.

报告人简介：

刁怀安，博士毕业于香港城市大学，东北师范大学数学与统计学院副教授，研究方向数值代数与反散射问题，在Mathematics of Computation, BIT, Numerical Linear Algebra with Applications, Linear Algebra and its Applications等国际知名期刊发表科研论文三十余篇；出版学术专著一本；曾主持国家自然科学基金青年基金项目1项，数学天元基金1项，教育部博士点新教师基金1项；现为吉林省工业与应用数学学会第四届理事会理事,国际线性代数协会会员；曾多次赴普渡大学、麦克马斯特大学、汉堡工业大学、日本国立信息研究所、香港科技大学、香港浸会大学等高校进行合作研究与学术访问。